Question

1-Prove the identity.

cosh(x) + sinh(x) = e^x

cosh(x) + sinh(x)	=	1 2 ex + e−x + 1 2
	=	1 2
	=

2-Prove the identity.

sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y)

sinh(x) cosh(y) + cosh(x) sinh(y)	=	1 2 (ex − e−x) 1 2    + 1 2 (ex + e−x) 1 2
	=	1 4   + ex − y − e−x + y − e−x − y + ex + y − ex − y + e−x + y − e−x − y
	=	1 4   − 2e−x − y
	=	1 2 ex + y − e−(x + y)
	=

3-Prove the identity.

sinh(2x) = 2 sinh(x) cosh(x)

sinh(2x)	=	sinh x +
	=	sinh(x)   + cosh(x) sinh(x)
	=

4-Prove the identity.

1 + tanh(x)

1 − tanh(x)

= e^2x

1 + tanh(x) 1 − tanh(x)	=	1 + cosh(x) 1 − cosh(x)
	=	cosh(x) + cosh(x) −
	=	1 2 ex + e−x + 1 2 (ex − e−x 1 2 ex + e−x − 1 2 (ex − e−x
	=	e−x
	=

5-If

tanh(x) =

4

5

,

find the values of the other hyperbolic functions at x.

sinh(x)	=
cosh(x)	=
coth(x)	=
sech(x)	=
csch(x)	=

Answer 1